Source code for thermosteam.properties.lennard_jones

# -*- coding: utf-8 -*-
# BioSTEAM: The Biorefinery Simulation and Techno-Economic Analysis Modules
# Copyright (C) 2020, Yoel Cortes-Pena <yoelcortes@gmail.com>
# 
# A significant portion of this module originates from:
# Chemical Engineering Design Library (ChEDL). Utilities for process modeling.
# Copyright (C) 2020 Caleb Bell <Caleb.Andrew.Bell@gmail.com>
# 
# This module is under a dual license:
# 1. The UIUC open-source license. See 
# github.com/BioSTEAMDevelopmentGroup/biosteam/blob/master/LICENSE.txt
# for license details.
# 
# 2. The MIT open-source license. See
# https://github.com/CalebBell/thermo/blob/master/LICENSE.txt for details.
'''
All data and functions for computing the Lennard-Jones molecular diameter.
'''

__all__ = ['MagalhaesLJ_data', 'Stockmayer_methods', 'Stockmayer', 
'molecular_diameter_methods', 'molecular_diameter', 'sigma_Flynn', 
'sigma_Bird_Stewart_Lightfoot_critical_2', 
'sigma_Bird_Stewart_Lightfoot_critical_1', 
'sigma_Bird_Stewart_Lightfoot_boiling', 'sigma_Bird_Stewart_Lightfoot_melting',
'sigma_Stiel_Thodos', 'sigma_Tee_Gotoh_Steward_1', 'sigma_Tee_Gotoh_Steward_2',
'sigma_Silva_Liu_Macedo', 'epsilon_Flynn', 
'epsilon_Bird_Stewart_Lightfoot_critical', 
'epsilon_Bird_Stewart_Lightfoot_boiling', 
'epsilon_Bird_Stewart_Lightfoot_melting', 'epsilon_Stiel_Thodos', 
'epsilon_Tee_Gotoh_Steward_1', 'epsilon_Tee_Gotoh_Steward_2', 
'Neufeld_collision', 'collision_integral_Neufeld_Janzen_Aziz', 'As_collision',
'Bs_collision', 'Cs_collision', 'collision_integral_Kim_Monroe', 'Tstar']

import os
import pandas as pd
from math import exp, log, sin
from .._constants import k

folder = os.path.join(os.path.dirname(__file__), 'Data/Viscosity')

MagalhaesLJ_data = pd.read_csv(os.path.join(folder,
                        'MagalhaesLJ.tsv'), sep='\t', index_col=0)


FLYNN = 'Flynn (1960)'
STIELTHODOS = 'Stiel and Thodos Tc, Zc (1962)'
MAGALHAES = 'Magalhães, Lito, Da Silva, and Silva (2013)'
TEEGOTOSTEWARD1 = 'Tee, Gotoh, and Stewart CSP with Tc (1966)'
TEEGOTOSTEWARD2 = 'Tee, Gotoh, and Stewart CSP with Tc, omega (1966)'
BSLC = 'Bird, Stewart, and Light (2002) critical relation'
BSLB = 'Bird, Stewart, and Light (2002) boiling relation'
BSLM = 'Bird, Stewart, and Light (2002) melting relation'
NONE = 'None'

Stockmayer_methods = [MAGALHAES, TEEGOTOSTEWARD2, FLYNN, BSLC, TEEGOTOSTEWARD1,
                      BSLB, BSLM, STIELTHODOS]


[docs]def Stockmayer(Tm=None, Tb=None, Tc=None, Zc=None, omega=None, CASRN='', AvailableMethods=False, Method=None): r'''This function handles the retrieval or calculation a chemical's Stockmayer parameter. Values are available from one source with lookup based on CASRNs, or can be estimated from 7 CSP methods. Will automatically select a data source to use if no Method is provided; returns None if the data is not available. Prefered sources are 'Magalhães, Lito, Da Silva, and Silva (2013)' for common chemicals which had valies listed in that source, and the CSP method `Tee, Gotoh, and Stewart CSP with Tc, omega (1966)` for chemicals which don't. Examples -------- >>> Stockmayer(CASRN='64-17-5') 1291.41 Parameters ---------- Tm : float, optional Melting temperature of fluid [K] Tb : float, optional Boiling temperature of fluid [K] Tc : float, optional Critical temperature, [K] Zc : float, optional Critical compressibility, [-] omega : float, optional Acentric factor of compound, [-] CASRN : string, optional CASRN [-] Returns ------- epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K] methods : list, only returned if AvailableMethods == True List of methods which can be used to obtain epsilon with the given inputs Other Parameters ---------------- Method : string, optional A string for the method name to use, as defined by constants in Stockmayer_methods AvailableMethods : bool, optional If True, function will determine which methods can be used to obtain epsilon for the desired chemical, and will return methods instead of epsilon Notes ----- These values are somewhat rough, as they attempt to pigeonhole a chemical into L-J behavior. The tabulated data is from [2]_, for 322 chemicals. References ---------- .. [1] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006 .. [2] Magalhães, Ana L., Patrícia F. Lito, Francisco A. Da Silva, and Carlos M. Silva. "Simple and Accurate Correlations for Diffusion Coefficients of Solutes in Liquids and Supercritical Fluids over Wide Ranges of Temperature and Density." The Journal of Supercritical Fluids 76 (April 2013): 94-114. doi:10.1016/j.supflu.2013.02.002. ''' def list_methods(): methods = [] if CASRN in MagalhaesLJ_data.index: methods.append(MAGALHAES) if Tc and omega: methods.append(TEEGOTOSTEWARD2) if Tc: methods.append(FLYNN) methods.append(BSLC) methods.append(TEEGOTOSTEWARD1) if Tb: methods.append(BSLB) if Tm: methods.append(BSLM) if Tc and Zc: methods.append(STIELTHODOS) methods.append(NONE) return methods if AvailableMethods: return list_methods() if not Method: Method = list_methods()[0] if Method == FLYNN: epsilon = epsilon_Flynn(Tc) elif Method == BSLC: epsilon = epsilon_Bird_Stewart_Lightfoot_critical(Tc) elif Method == BSLB: epsilon = epsilon_Bird_Stewart_Lightfoot_boiling(Tb) elif Method == BSLM: epsilon = epsilon_Bird_Stewart_Lightfoot_melting(Tm) elif Method == STIELTHODOS: epsilon = epsilon_Stiel_Thodos(Tc, Zc) elif Method == TEEGOTOSTEWARD1: epsilon = epsilon_Tee_Gotoh_Steward_1(Tc) elif Method == TEEGOTOSTEWARD2: epsilon = epsilon_Tee_Gotoh_Steward_2(Tc, omega) elif Method == MAGALHAES: epsilon = float(MagalhaesLJ_data.at[CASRN, "epsilon"]) elif Method == NONE: epsilon = None else: raise Exception('Failure in in function') return epsilon
TEEGOTOSTEWARD3 = 'Tee, Gotoh, and Stewart CSP with Tc, Pc (1966)' TEEGOTOSTEWARD4 = 'Tee, Gotoh, and Stewart CSP with Tc, Pc, omega (1966)' BSLC1 = 'Bird, Stewart, and Light (2002) critical relation with Vc' BSLC2 = 'Bird, Stewart, and Light (2002) critical relation with Tc, Pc' STIELTHODOSMD = 'Stiel and Thodos Vc, Zc (1962)' SILVALIUMACEDO = 'Silva, Liu, and Macedo (1998) critical relation with Tc, Pc' molecular_diameter_methods = [MAGALHAES, TEEGOTOSTEWARD4, SILVALIUMACEDO, BSLC2, TEEGOTOSTEWARD3, STIELTHODOSMD, FLYNN, BSLC1, BSLB, BSLM]
[docs]def molecular_diameter(Tc=None, Pc=None, Vc=None, Zc=None, omega=None, Vm=None, Vb=None, CASRN='', AvailableMethods=False, Method=None): r'''This function handles the retrieval or calculation a chemical's L-J molecular diameter. Values are available from one source with lookup based on CASRNs, or can be estimated from 9 CSP methods. Will automatically select a data source to use if no Method is provided; returns None if the data is not available. Prefered sources are 'Magalhães, Lito, Da Silva, and Silva (2013)' for common chemicals which had valies listed in that source, and the CSP method `Tee, Gotoh, and Stewart CSP with Tc, Pc, omega (1966)` for chemicals which don't. Examples -------- >>> molecular_diameter(CASRN='64-17-5') 4.23738 Parameters ---------- Tc : float, optional Critical temperature, [K] Pc : float, optional Critical pressure, [Pa] Vc : float, optional Critical volume, [m^3/mol] Zc : float, optional Critical compressibility, [-] omega : float, optional Acentric factor of compound, [-] Vm : float, optional Molar volume of liquid at the melting point of the fluid [K] Vb : float, optional Molar volume of liquid at the boiling point of the fluid [K] CASRN : string, optional CASRN [-] Returns ------- sigma : float Lennard-Jones molecular diameter, [Angstrom] methods : list, only returned if AvailableMethods == True List of methods which can be used to obtain epsilon with the given inputs Other Parameters ---------------- Method : string, optional A string for the method name to use, as defined by constants in molecular_diameter_methods AvailableMethods : bool, optional If True, function will determine which methods can be used to obtain sigma for the desired chemical, and will return methods instead of sigma Notes ----- These values are somewhat rough, as they attempt to pigeonhole a chemical into L-J behavior. The tabulated data is from [2]_, for 322 chemicals. References ---------- .. [1] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006 .. [2] Magalhães, Ana L., Patrícia F. Lito, Francisco A. Da Silva, and Carlos M. Silva. "Simple and Accurate Correlations for Diffusion Coefficients of Solutes in Liquids and Supercritical Fluids over Wide Ranges of Temperature and Density." The Journal of Supercritical Fluids 76 (April 2013): 94-114. doi:10.1016/j.supflu.2013.02.002. ''' def list_methods(): methods = [] if CASRN in MagalhaesLJ_data.index: methods.append(MAGALHAES) if Tc and Pc and omega: methods.append(TEEGOTOSTEWARD4) if Tc and Pc: methods.append(SILVALIUMACEDO) methods.append(BSLC2) methods.append(TEEGOTOSTEWARD3) if Vc and Zc: methods.append(STIELTHODOSMD) if Vc: methods.append(FLYNN) methods.append(BSLC1) if Vb: methods.append(BSLB) if Vm: methods.append(BSLM) methods.append(NONE) return methods if AvailableMethods: return list_methods() if not Method: Method = list_methods()[0] if Method == FLYNN: sigma = sigma_Flynn(Vc) elif Method == BSLC1: sigma = sigma_Bird_Stewart_Lightfoot_critical_1(Vc) elif Method == BSLC2: sigma = sigma_Bird_Stewart_Lightfoot_critical_2(Tc, Pc) elif Method == TEEGOTOSTEWARD3: sigma = sigma_Tee_Gotoh_Steward_1(Tc, Pc) elif Method == SILVALIUMACEDO: sigma = sigma_Silva_Liu_Macedo(Tc, Pc) elif Method == BSLB: sigma = sigma_Bird_Stewart_Lightfoot_boiling(Vb) elif Method == BSLM: sigma = sigma_Bird_Stewart_Lightfoot_melting(Vm) elif Method == STIELTHODOSMD: sigma = sigma_Stiel_Thodos(Vc, Zc) elif Method == TEEGOTOSTEWARD4: sigma = sigma_Tee_Gotoh_Steward_2(Tc, Pc, omega) elif Method == MAGALHAES: sigma = float(MagalhaesLJ_data.at[CASRN, "sigma"]) elif Method == NONE: sigma = None else: raise Exception('Failure in in function') return sigma
### Sigma Lennard-Jones
[docs]def sigma_Flynn(Vc): r'''Calculates Lennard-Jones molecular diameter. Uses critical volume. CSP method by [1]_ as reported in [2]_. .. math:: \sigma = 0.561(V_c^{1/3})^{5/4} Parameters ---------- Vc : float Critical volume of fluid [m^3/mol] Returns ------- sigma : float Lennard-Jones molecular diameter, [Angstrom] Notes ----- Vc is originally in units of mL/mol. Examples -------- >>> sigma_Flynn(0.000268) 5.2506948422196285 References ---------- .. [1] Flynn, L.W., M.S. thesis, Northwestern Univ., Evanston, Ill. (1960). .. [2] Stiel, L. I., and George Thodos. "Lennard-Jones Force Constants Predicted from Critical Properties." Journal of Chemical & Engineering Data 7, no. 2 (April 1, 1962): 234-36. doi:10.1021/je60013a023 ''' Vc = Vc*1E6 # m^3/mol to cm^3/mol sigma = 0.561*(Vc**(1/3.))**1.2 return sigma
[docs]def sigma_Bird_Stewart_Lightfoot_critical_2(Tc, Pc): r'''Calculates Lennard-Jones molecular diameter. Uses critical temperature and pressure. CSP method by [1]_. .. math:: \sigma = 2.44(T_c/P_c)^{1/3} Parameters ---------- Tc : float Critical temperature of fluid [K] Pc : float Critical pressure of fluid [Pa] Returns ------- sigma : float Lennard-Jones molecular diameter, [Angstrom] Notes ----- Original units of critical pressure are atmospheres. Examples -------- >>> sigma_Bird_Stewart_Lightfoot_critical_2(560.1, 4550000) 5.658657684653222 References ---------- .. [1] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006 ''' Pc = Pc/101325. sigma = 2.44*(Tc/Pc)**(1/3.0) return sigma
[docs]def sigma_Bird_Stewart_Lightfoot_critical_1(Vc): r'''Calculates Lennard-Jones molecular diameter. Uses critical volume. CSP method by [1]_. .. math:: \sigma = 0.841 V_c^{1/3} Parameters ---------- Vc : float Critical volume of fluid [m^3/mol] Returns ------- sigma : float Lennard-Jones molecular diameter, [Angstrom] Notes ----- Original units of Vc are mL/mol. Examples -------- >>> sigma_Bird_Stewart_Lightfoot_critical_1(0.000268) 5.422184116631474 References ---------- .. [1] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006 ''' Vc = Vc*1E6 # m^3/mol to cm^3/mol sigma = 0.841*Vc**(1/3.0) return sigma
[docs]def sigma_Bird_Stewart_Lightfoot_boiling(Vb): r'''Calculates Lennard-Jones molecular diameter. Uses molar volume of liquid at boiling. CSP method by [1]_. .. math:: \sigma = 1.166V_{b,liq}^{1/3} Parameters ---------- Vb : float Boiling molar volume of liquid [m^3/mol] Returns ------- sigma : float Lennard-Jones collision integral, [Angstrom] Notes ----- Original units of Vb are mL/mol. Examples -------- >>> sigma_Bird_Stewart_Lightfoot_boiling(0.0001015) 5.439018856944655 References ---------- .. [1] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006 ''' Vb = Vb*1E6 sigma = 1.166*Vb**(1/3.0) return sigma
[docs]def sigma_Bird_Stewart_Lightfoot_melting(Vm): r'''Calculates Lennard-Jones molecular diameter. Uses molar volume of a liquid at its melting point. CSP method by [1]_. .. math:: \sigma = 1.222 V_{m,sol}^{1/3} Parameters ---------- Vm : float Melting molar volume of a liquid at its melting point [m^3/mol] Returns ------- sigma : float Lennard-Jones molecular diameter, [Angstrom] Notes ----- Original units of Vm are mL/mol. Examples -------- >>> sigma_Bird_Stewart_Lightfoot_melting(8.8e-05) 5.435407341351406 References ---------- .. [1] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006 ''' Vm = Vm*1E6 sigma = 1.222*Vm**(1/3.) return sigma
[docs]def sigma_Stiel_Thodos(Vc, Zc): r'''Calculates Lennard-Jones molecular diameter. Uses critical volume and compressibility. CSP method by [1]_. .. math:: \sigma = 0.1866 V_c^{1/3} Z_c^{-6/5} Parameters ---------- Vc : float Critical volume of fluid [m^3/mol] Zc : float Critical compressibility of fluid, [-] Returns ------- sigma : float Lennard-Jones molecular diameter, [Angstrom] Notes ----- Vc is originally in units of mL/mol. Examples -------- Monofluorobenzene >>> sigma_Stiel_Thodos(0.000271, 0.265) 5.94300853971033 References ---------- .. [1] Stiel, L. I., and George Thodos. "Lennard-Jones Force Constants Predicted from Critical Properties." Journal of Chemical & Engineering Data 7, no. 2 (April 1, 1962): 234-36. doi:10.1021/je60013a023 ''' Vc = Vc*1E6 sigma = 0.1866*Vc**(1/3.0)*Zc**(-1.2) return sigma
[docs]def sigma_Tee_Gotoh_Steward_1(Tc, Pc): r'''Calculates Lennard-Jones molecular diameter. Uses critical temperature and pressure. CSP method by [1]_. .. math:: \sigma = 2.3647 \left(\frac{T_c}{P_c}\right)^{1/3} Parameters ---------- Tc : float Critical temperature of fluid [K] Pc : float Critical pressure of fluid [Pa] Returns ------- sigma : float Lennard-Jones molecular diameter, [Angstrom] Notes ----- Original units of Pc are atm. Further regressions with other parameters were performed in [1]_ but are not included here, except for `sigma_Tee_Gotoh_Steward_2`. Examples -------- >>> sigma_Tee_Gotoh_Steward_1(560.1, 4550000) 5.48402779790962 References ---------- .. [1] Tee, L. S., Sukehiro Gotoh, and W. E. Stewart. "Molecular Parameters for Normal Fluids. Lennard-Jones 12-6 Potential." Industrial & Engineering Chemistry Fundamentals 5, no. 3 (August 1, 1966): 356-63. doi:10.1021/i160019a011 ''' Pc = Pc/101325. sigma = 2.3647*(Tc/Pc)**(1/3.) return sigma
[docs]def sigma_Tee_Gotoh_Steward_2(Tc, Pc, omega): r'''Calculates Lennard-Jones molecular diameter. Uses critical temperature, pressure, and acentric factor. CSP method by [1]_. .. math:: \sigma = (2.3551 - 0.0874\omega)\left(\frac{T_c}{P_c}\right)^{1/3} Parameters ---------- Tc : float Critical temperature of fluid [K] Pc : float Critical pressure of fluid [Pa] omega : float Acentric factor for fluid, [-] Returns ------- sigma : float Lennard-Jones molecular diameter, [Angstrom] Notes ----- Original units of Pc are atm. Further regressions with other parameters were performed in [1]_ but are not included here, except for `sigma_Tee_Gotoh_Steward_1`. Examples -------- >>> sigma_Tee_Gotoh_Steward_2(560.1, 4550000, 0.245) 5.412104867264477 References ---------- .. [1] Tee, L. S., Sukehiro Gotoh, and W. E. Stewart. "Molecular Parameters for Normal Fluids. Lennard-Jones 12-6 Potential." Industrial & Engineering Chemistry Fundamentals 5, no. 3 (August 1, 1966): 356-63. doi:10.1021/i160019a011 ''' Pc = Pc/101325. sigma = (2.3551-0.0874*omega)*(Tc/Pc)**(1/3.) return sigma
[docs]def sigma_Silva_Liu_Macedo(Tc, Pc): r'''Calculates Lennard-Jones molecular diameter. Uses critical temperature and pressure. CSP method by [1]_. .. math:: \sigma_{LJ}^3 = 0.17791 + 11.779 \left( \frac{T_c}{P_c}\right) - 0.049029\left( \frac{T_c}{P_c}\right)^2 Parameters ---------- Tc : float Critical temperature of fluid [K] Pc : float Critical pressure of fluid [Pa] Returns ------- sigma : float Lennard-Jones molecular diameter, [Angstrom] Notes ----- Pc is originally in bar. An excellent paper. None is returned if the polynomial returns a negative number, as in the case of 1029.13 K and 3.83 bar. Examples -------- >>> sigma_Silva_Liu_Macedo(560.1, 4550000) 5.164483998730177 References ---------- .. [1] Silva, Carlos M., Hongqin Liu, and Eugenia A. Macedo. "Models for Self-Diffusion Coefficients of Dense Fluids, Including Hydrogen-Bonding Substances." Chemical Engineering Science 53, no. 13 (July 1, 1998): 2423-29. doi:10.1016/S0009-2509(98)00037-2 ''' Pc = Pc/1E5 # Pa to bar term = 0.17791 + 11.779*(Tc/Pc) - 0.049029 * (Tc/Pc)**2 if term < 0: sigma = None else: sigma = (term)**(1/3.) return sigma
### epsilon Lennard-Jones
[docs]def epsilon_Flynn(Tc): r'''Calculates Lennard-Jones depth of potential-energy minimum. Uses critical temperature. CSP method by [1]_ as reported in [2]_. .. math:: \epsilon/k = 1.77 T_c^{5/6} Parameters ---------- Tc : float Critical temperature of fluid [K] Returns ------- epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K] Notes ----- Examples -------- >>> epsilon_Flynn(560.1) 345.2984087011443 References ---------- .. [1] Flynn, L.W., M.S. thesis, Northwestern Univ., Evanston, Ill. (1960). .. [2] Stiel, L. I., and George Thodos. "Lennard-Jones Force Constants Predicted from Critical Properties." Journal of Chemical & Engineering Data 7, no. 2 (April 1, 1962): 234-36. doi:10.1021/je60013a023 ''' epsilon_k = 1.77*Tc**(5/6.) return epsilon_k
[docs]def epsilon_Bird_Stewart_Lightfoot_critical(Tc): r'''Calculates Lennard-Jones depth of potential-energy minimum. Uses critical temperature. CSP method by [1]_. .. math:: \epsilon/k = 0.77T_c Parameters ---------- Tc : float Critical temperature of fluid [K] Returns ------- epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K] Notes ----- Examples -------- >>> epsilon_Bird_Stewart_Lightfoot_critical(560.1) 431.27700000000004 References ---------- .. [1] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006 ''' epsilon_k = 0.77*Tc return epsilon_k
[docs]def epsilon_Bird_Stewart_Lightfoot_boiling(Tb): r'''Calculates Lennard-Jones depth of potential-energy minimum. Uses boiling temperature. CSP method by [1]_. .. math:: \epsilon/k = 1.15 T_b Parameters ---------- Tb : float Boiling temperature [K] Returns ------- epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K] Notes ----- Examples -------- >>> epsilon_Bird_Stewart_Lightfoot_boiling(357.85) 411.5275 References ---------- .. [1] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006 ''' epsilon_k = 1.15*Tb return epsilon_k
[docs]def epsilon_Bird_Stewart_Lightfoot_melting(Tm): r'''Calculates Lennard-Jones depth of potential-energy minimum. Uses melting temperature. CSP method by [1]_. .. math:: \epsilon/k = 1.92T_m Parameters ---------- Tm : float Melting temperature [K] Returns ------- epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K] Notes ----- Examples -------- >>> epsilon_Bird_Stewart_Lightfoot_melting(231.15) 443.808 References ---------- .. [1] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006 ''' epsilon_k = 1.92*Tm return epsilon_k
[docs]def epsilon_Stiel_Thodos(Tc, Zc): r'''Calculates Lennard-Jones depth of potential-energy minimum. Uses Critical temperature and critical compressibility. CSP method by [1]_. .. math:: \epsilon/k = 65.3 T_c Z_c^{3.6} Parameters ---------- Tc : float Critical temperature of fluid [K] Zc : float Critical compressibility of fluid, [-] Returns ------- epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K] Notes ----- Examples -------- Fluorobenzene >>> epsilon_Stiel_Thodos(358.5, 0.265) 196.3755830305783 References ---------- .. [1] Stiel, L. I., and George Thodos. "Lennard-Jones Force Constants Predicted from Critical Properties." Journal of Chemical & Engineering Data 7, no. 2 (April 1, 1962): 234-36. doi:10.1021/je60013a023 ''' epsilon_k = 65.3*Tc*Zc**3.6 return epsilon_k
[docs]def epsilon_Tee_Gotoh_Steward_1(Tc): r'''Calculates Lennard-Jones depth of potential-energy minimum. Uses Critical temperature. CSP method by [1]_. .. math:: \epsilon/k = 0.7740T_c Parameters ---------- Tc : float Critical temperature of fluid [K] Returns ------- epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K] Notes ----- Further regressions with other parameters were performed in [1]_ but are not included here, except for `epsilon_Tee_Gotoh_Steward_2`. Examples -------- >>> epsilon_Tee_Gotoh_Steward_1(560.1) 433.5174 References ---------- .. [1] Tee, L. S., Sukehiro Gotoh, and W. E. Stewart. "Molecular Parameters for Normal Fluids. Lennard-Jones 12-6 Potential." Industrial & Engineering Chemistry Fundamentals 5, no. 3 (August 1, 1966): 356-63. doi:10.1021/i160019a011 ''' epsilon_k = 0.7740*Tc return epsilon_k
[docs]def epsilon_Tee_Gotoh_Steward_2(Tc, omega): r'''Calculates Lennard-Jones depth of potential-energy minimum. Uses critical temperature and acentric factor. CSP method by [1]_. .. math:: \epsilon/k = (0.7915 + 0.1693 \omega)T_c Parameters ---------- Tc : float Critical temperature of fluid [K] omega : float Acentric factor for fluid, [-] Returns ------- epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K] Notes ----- Further regressions with other parameters were performed in [1]_ but are not included here, except for `epsilon_Tee_Gotoh_Steward_1`. Examples -------- >>> epsilon_Tee_Gotoh_Steward_2(560.1, 0.245) 466.55125785 References ---------- .. [1] Tee, L. S., Sukehiro Gotoh, and W. E. Stewart. "Molecular Parameters for Normal Fluids. Lennard-Jones 12-6 Potential." Industrial & Engineering Chemistry Fundamentals 5, no. 3 (August 1, 1966): 356-63. doi:10.1021/i160019a011 ''' epsilon_k = (0.7915 + 0.1693*omega)*Tc return epsilon_k
### Collision Integral Neufeld_collision = { (1, 1): [1.06036, 0.1561, 0.193, 0.47635, 1.03587, 1.52996, 1.76474, 3.89411, None, None, None, None], (1, 2): [1.0022, 0.1553, 0.16105, 0.72751, 0.86125, 2.06848, 1.95162, 4.84492, None, None, None, None], (1, 3): [0.96573, 0.15611, 0.44067, 1.5242, 2.38981, 5.08063, None, None, -0.0005373, 19.2866, -1.30775, 6.58711], (1, 4): [0.93447, 0.15578, 0.39478, 1.85761, 2.45988, 6.15727, None, None, 0.0004246, 12.988, -1.36399, 3.3329], (1, 5): [0.90972, 0.15565, 0.35967, 2.18528, 2.45169, 7.17936, None, None, -0.0003814, 9.38191, 0.14025, 9.93802], (1, 6): [0.88928, 0.15562, 0.33305, 2.51303, 2.36298, 8.1169, None, None, -0.0004649, 9.86928, 0.12851, 9.82414], (1, 7): [0.87208, 0.15568, 0.36583, 3.01399, 2.70659, 9.9231, None, None, -0.0004902, 10.2274, 0.12306, 9.97712], (2, 2): [1.16145, 0.14874, 0.52487, 0.7732, 2.16178, 2.43787, None, None, -0.0006435, 18.0323, -0.7683, 7.27371], (2, 3): [1.11521, 0.14796, 0.44844, 0.99548, 2.30009, 3.06031, None, None, 0.0004565, 38.5868, -0.69403, 2.56375], (2, 4): [1.08228, 0.14807, 0.47128, 1.31596, 2.42738, 3.90018, None, None, -0.0005623, 3.08449, 0.28271, 3.22871], (2, 5): [1.05581, 0.14822, 0.51203, 1.67007, 2.57317, 4.85939, None, None, -0.000712, 4.7121, 0.2173, 4.7353], (2, 6): [1.03358, 0.14834, 0.53928, 2.01942, 2.7235, 5.84817, None, None, -0.0008576, 7.66012, 0.15493, 7.6011], (3, 3): [1.05567, 0.1498, 0.30887, 0.86437, 1.35766, 2.44123, 1.2903, 5.55734, 0.0002339, 57.7757, -1.0898, 6.9475], (3, 4): [1.02621, 0.1505, 0.55381, 1.4007, 2.06176, 4.26234, None, None, 0.0005227, 11.3331, -0.8209, 3.87185], (3, 5): [0.99958, 0.15029, 0.50441, 1.64304, 2.06947, 4.87712, None, None, -0.0005184, 3.45031, 0.26821, 3.73348], (4, 4): [1.12007, 0.14578, 0.53347, 1.11986, 2.28803, 3.27567, None, None, 0.0007427, 21.048, -0.28759, 6.69149] }
[docs]def collision_integral_Neufeld_Janzen_Aziz(Tstar, l=1, s=1): r'''Calculates Lennard-Jones collision integral for any of 16 values of (l,j) for the wide range of 0.3 < Tstar < 100. Values are accurate to 0.1 % of actual values, but the calculation of actual values is computationally intensive and so these simplifications are used, developed in [1]_. .. math:: \Omega_D = \frac{A}{T^{*B}} + \frac{C}{\exp(DT^*)} + \frac{E}{\exp(FT^{*})} + \frac{G}{\exp(HT^*)} + RT^{*B}\sin(ST^{*W}-P) Parameters ---------- Tstar : float Reduced temperature of the fluid [-] l : int term s : int term Returns ------- Omega : float Collision integral of A and B Notes ----- Acceptable pairs of (l,s) are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), and (4, 4). .. math:: T^* = \frac{k_b T}{\epsilon} Results are very similar to those of the more modern formulation, `collision_integral_Kim_Monroe`. Calculations begin to yield overflow errors in some values of (l, 2) after Tstar = 75, beginning with (1, 7). Also susceptible are (1, 5) and (1, 6). Examples -------- >>> collision_integral_Neufeld_Janzen_Aziz(100, 1, 1) 0.516717697672334 References ---------- .. [1] Neufeld, Philip D., A. R. Janzen, and R. A. Aziz. "Empirical Equations to Calculate 16 of the Transport Collision Integrals Omega(l, S)* for the Lennard-Jones (12-6) Potential." The Journal of Chemical Physics 57, no. 3 (August 1, 1972): 1100-1102. doi:10.1063/1.1678363 ''' if (l, s) not in Neufeld_collision: raise Exception('Input values of l and s are not supported') A, B, C, D, E, F, G, H, R, S, W, P = Neufeld_collision[(l, s)] omega = A/Tstar**B + C/exp(D*Tstar) + E/exp(F*Tstar) if (l, s) in [(1, 1), (1, 2), (3, 3)]: omega += G/exp(H*Tstar) if (l, s) not in [(1, 1), (1, 2)]: omega += R*Tstar**B*sin(S*Tstar**W-P) return omega
As_collision = {(1, 1): -1.10367290, (1, 2): 1.35555540, (1, 3): 1.06771150, (1, 4): 0.80959899, (1, 5): 0.74128322, (1, 6): 0.80998324, (1, 7): 0.81808091, (2, 2): -0.92032979, (2, 3): 2.59557990, (2, 4): 1.60427450, (2, 5): 0.82064641, (2, 6): 0.79413652, (3, 3): 1.26304910, (3, 4): 2.21146360, (3, 5): 1.50498090, (4, 4): 2.62223930 } Bs_collision = { (1, 1): [2.6431984,0.0060432255,-0.15158773,0.054237938,-0.0090468682,0.0006174200700], (1, 2): [-0.44668594,0.42734391,-0.16036459,0.031461648,-0.0032587575,0.0001386025700], (1, 3): [-0.1394539,0.17696362,-0.026252211,-0.0043814141,0.00167521,-0.0001438280100], (1, 4): [0.1293817,0.059760309,0.0071109469,-0.0063851124,0.0010498938,-0.0000581492570], (1, 5): [0.1778885,0.027398438,0.0076254248,-0.0031650182,0.0003278652,-0.0000092890016], (1, 6): [0.073071217,0.034607908,-0.0011457199,0.000281986,-0.0002006054,0.0000214464830], (1, 7): [0.044232851,0.029750283,-0.0022011682,0.0006326412,-0.0001755553,0.0000142557040], (2, 2): [2.3508044,0.50110649,-0.47193769,0.15806367,-0.026367184,0.0018120118000], (2, 3): [-1.8569443,0.96985775,-0.39888526,0.090063692,-0.010918991,0.0005664679700], (2, 4): [-0.67406115,0.42671907,-0.10177069,0.0006185714,0.0031225358,-0.0003520605100], (2, 5): [0.23195128,0.12233793,0.013891578,-0.020903423,0.0046715462,-0.0003520430300], (2, 6): [0.23766123,0.077125802,0.013060901,-0.010982362,0.0018034505,-0.0000959825710], (3, 3): [-0.36104243,0.68116214,-0.36401583,0.10500196,-0.016400134,0.0010880886000], (3, 4): [-1.4743107,0.64918549,-0.24075196,0.051820149,-0.0060565396,0.0002981232600], (3, 5): [-0.64335529,0.3261704,-0.082126072,0.0059682011,0.0010269488,-0.0001595725200], (4, 4): [-1.9158462,1.016638,-0.43355278,0.10496591,-0.013951104,0.0008004853400] } Cs_collision = { (1, 1): [1.6690746, -0.6914589, 0.15502132, -0.020642189, 0.001540207700, -0.000049729535], (1, 2): [-0.47499422, 0.14482036, -0.032158368, 0.0044357933, -0.00034138118, 0.000011259742], (1, 3): [-0.25258689, 0.059709197, -0.013332695, 0.0019619285, -0.000160630760, 0.0000055804557], (1, 4): [-0.045055948, -0.022642753, 0.0056672308, -0.0006570876, 0.000040733113, -0.0000010820157], (1, 5): [0.0013668724, -0.041730962, 0.010378923, -0.0013492954, 0.000096963599, -0.0000030307552], (1, 6): [-0.071180849, -0.012738119, 0.0038582834, -0.0004706043, 0.000030466929, -0.00000085305576], (1, 7): [-0.089417548, -0.0051856424, 0.0021882143, -0.0002487447, 0.000013745859, -0.00000030285365], (2, 2): [1.6330213, -0.69795156, 0.16096572, -0.02210944, 0.0017031434, -0.000056699986], (2, 3): [-1.4586197, 0.52947262, -0.11946363, 0.016264589, -0.0012354315, 0.000040366357], (2, 4): [-0.62774499, 0.20700644, -0.04760169, 0.0067153792, -0.00052706167, 0.000017705708], (2, 5): [0.039184885, -0.057316906, 0.012794497, -0.0015336449, 0.00010241454, -0.0000029975563], (2, 6): [0.050470266, -0.062621672, 0.014326724, -0.0017806541, 0.00012353365, -0.0000037501381], (3, 3): [-0.33227158, 0.079723851, -0.015470355, 0.0018686705, -0.00012179945, 0.0000032594587], (3, 4): [-1.1942554, 0.43000688, -0.097525871, 0.013399366, -0.0010283777, 0.000033956674], (3, 5): [-0.60014514, 0.19764859, -0.045212434, 0.0063650284, -0.00049991689, 0.000016833944], (4, 4): [-1.4676253, 0.53048161, -0.11909781, 0.016123847, -0.0012174905, 0.0000395451] }
[docs]def collision_integral_Kim_Monroe(Tstar, l=1, s=1): r'''Calculates Lennard-Jones collision integral for any of 16 values of (l,j) for the wide range of 0.3 < Tstar < 400. Values are accurate to 0.007 % of actual values, but the calculation of actual values is computationally intensive and so these simplifications are used, developed in [1]_. .. math:: \Omega^{(l,s)*} = A^{(l,s)} + \sum_{k=1}^6 \left[ \frac{B_k^{(l,s)}} {(T^*)^k} + C_k^{(l,s)} (\ln T^*)^k \right] Parameters ---------- Tstar : float Reduced temperature of the fluid [-] l : int term s : int term Returns ------- Omega : float Collision integral of A and B Notes ----- Acceptable pairs of (l,s) are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), and (4, 4). .. math:: T^* = \frac{k_b T}{\epsilon} Examples -------- >>> collision_integral_Kim_Monroe(400, 1, 1) 0.4141818082392228 References ---------- .. [1] Kim, Sun Ung, and Charles W. Monroe. "High-Accuracy Calculations of Sixteen Collision Integrals for Lennard-Jones (12-6) Gases and Their Interpolation to Parameterize Neon, Argon, and Krypton." Journal of Computational Physics 273 (September 15, 2014): 358-73. doi:10.1016/j.jcp.2014.05.018. ''' if (l, s) not in As_collision: raise Exception('Input values of l and s are not supported') omega = As_collision[(l, s)] for ki in range(6): Bs = Bs_collision[(l, s)] Cs = Cs_collision[(l, s)] omega += Bs[ki]/Tstar**(ki+1) + Cs[ki]*log(Tstar)**(ki+1) return omega
### Misc
[docs]def Tstar(T, epsilon_k=None, epsilon=None): r'''This function calculates the parameter `Tstar` as needed in performing collision integral calculations. .. math:: T^* = \frac{kT}{\epsilon} Examples -------- >>> Tstar(T=318.2, epsilon_k=308.43) 1.0316765554582887 Parameters ---------- epsilon_k : float, optional Lennard-Jones depth of potential-energy minimum over k, [K] epsilon : float, optional Lennard-Jones depth of potential-energy minimum [J] Returns ------- Tstar : float Dimentionless temperature for calculating collision integral, [-] Notes ----- Tabulated values are normally listed as epsilon/k. k is the Boltzman constant, with units of J/K. References ---------- .. [1] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006 ''' if epsilon_k: _Tstar = T/(epsilon_k) elif epsilon: _Tstar = k*T/epsilon else: raise Exception('Either epsilon/k or epsilon must be provided') return _Tstar